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arXiv:cond-mat/0605183 · Physical Review A

Detecting phonons and persistent currents in toroidal Bose-Einstein condensates by means of pattern formation

Michele Modugno, C. Tozzo, F. Dalfovo

Published Dec 5, 2006• 68 citations
arXivHTMLPDF

Abstract

We theoretically investigate the dynamic properties of a Bose-Einstein condensate in a toroidal trap. A periodic modulation of the transverse confinement is shown to produce a density pattern due to parametric amplification of phonon pairs. By imaging the density distribution after free expansion one obtains (i) a precise determination of the Bogoliubov spectrum and (ii) a sensitive detection of quantized circulation in the torus. The parametric amplification is also sensitive to thermal and quantum fluctuations.

DOI
10.1103/physreva.74.061601

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We theoretically investigate the dynamic properties of a Bose-Einstein condensate in a toroidal trap. A periodic modulation of the transverse confinement is shown to produce a density pattern due to parametric amplification of phonon pairs. By imaging the density distribution after free expansion one obtains (i) a precise determination of the Bogoliubov spectrum and (ii) a sensitive detection of quantized circulation in the torus. The parametric amplification is also sensitive to thermal and quantum fluctuations.
We present a detailed investigation of one of the cleanest examples where it is possible to detect the “analog” Dynamical Casimir Effect in a Bose–Einstein condensate: an ultracold atom gas in toroidal confinement. The analytical solution of the time dependent Gross–Pitaevskii equation allows to follow the time evolution of the phonon spectrum and shows that periodic oscillations of the ring radius do not induce modulations in the density profile but give rise to the mixing of clockwise and anticlockwise modes, leading to the creation of pairs of entangled phonons in a squeezed vacuum state, if the drive frequency equals twice the frequency of the phonon mode. The Dynamical Casimir Effect is predicted to occur in the weakly interacting regime, where the Gross–Pitaevskii equation provides a faithful description of the many body dynamics. In the strong coupling limit, when the ultracold gas behaves as hard core bosons, the effect disappears and no amplification occurs. The presence of symmetry-breaking perturbations and finite temperature effects are also considered, as well as the comparison with the classical phenomenon of parametric amplification.
Superfluid properties of Bose-Einstein condensates (BEC) in toroidal quasi-one-dimensional traps are investigated in the presence of periodic scattering length modulations along the ring. The existence of several types of stable periodic waves, ranging from almost uniform to very fragmented chains of weakly interacting and equally spaced solitons, is demonstrated. We show that these waves may support persistent atomic currents and sound waves with spectra of Bogoliubov type. Fragmented condensates can be viewed as arrays of Josephson junctions and the current as a BEC manifestation of the dc-Josephson effect. The influence of linear defects on BEC superfluidity has been also investigated. We found that for subcritical velocities, linear defects that are static with respect to the lattice (while the condensate moves in respect to both the optical lattice and the defect) preserve the BEC superfluidity.
The dynamics of toroidal condensates in the presence of condensate flow and dipole perturbation have been investigated. The Bogoliubov spectrum of a condensate is calculated for an oblate torus using a discrete-variable representation and a spectral method to high accuracy. The transition from spheroidal to toroidal geometry of the trap displaces the energy levels into narrow bands. The lowest-order acoustic modes are quantized with the dispersion relation $\ensuremath{\omega}\ensuremath{\sim}|m|{\ensuremath{\omega}}_{s}$ with $m=0,\ifmmode\pm\else\textpm\fi{}1,\ifmmode\pm\else\textpm\fi{}2,\dots{}.$ A condensate with toroidal current $\ensuremath{\kappa}$ splits the ∣m∣|m|∣m∣ co-rotating and counter-rotating pairs by the amount $\ensuremath{\Delta}E\ensuremath{\approx}2|m|{\ensuremath{\Elzxh}}^{2}\ensuremath{\kappa}〈{r}^{\ensuremath{-}2}〉.$ Radial dipole excitations are the lowest-energy dissipation modes. For highly occupied condensates the nonlinearity creates an asymmetric mix of dipole circulation and nonlinear shifts in the spectrum of excitations so that the center of mass circulates around the axis of symmetry of the trap. We outline an experimental method to study these excitations.
We investigate rotational properties of a system of attractive bosons confined in a one-dimensional torus. Two kinds of ground states, uniform-density and bright soliton, are obtained analytically as functions of the strength of interaction and of the rotational frequency of the torus. The quantization of circulation appears in the uniform-density state, but disappears upon formation of the soliton. By comparing the results of exact diagonalization with those predicted by the Bogoliubov theory, we show that the Bogoliubov theory is valid at absolute zero over a wide range of parameters. At finite temperatures we employ the exact diagonalization method to examine how thermal fluctuations smear the plateaus of the quantized circulation. Finally, by rotating the system with an axisymmetry-breaking potential, we clarify the process by which the quantized circulation becomes thermodynamically stabilized.
We consider Bose-Einstein-condensed atoms confined in a toroidal trap. Under conditions of one-dimensional behavior, the density distribution of the atoms may be exponentially localized or delocalized, even for very small variations in the trapping potential along the torus. For similar reasons, small irregularities of the trap may also have a very pronounced effect on the density of the cloud. The excitation spectrum is also affected by the presence of some spatially varying potential and/or irregularities in the trap.
The radial trap frequency of an elongated harmonic trap containing a Bose-Einstein condensate is modulated. The free evolution of the condensate fraction is studied using phase-contrast imaging, revealing the dynamics of a single sample over a period of time. Multiple measurement series are combined to study the long-term behavior of samples created under similar experimental conditions. A high-order axial mode, the Faraday pattern, is observed. A simple parametric oscillator model is presented to describe the formation of this axial mode. This model suggests the radial quadrupole mode couples into the high-order axial mode through a parametric resonance. Also, the Faraday pattern is used to determine the speed of sound in the condensate as a function of density. The determined speed of sound deviates up to 20% from the Bogoliubov speed of sound for found densities.